For what follows, we need to introduce few useful concepts from [1]}.
For \(S\subset V\) , let the Hamming graph \({\mathcal {H}}_S\)  be the graph whose vertices correspond to
the configurations \(\lbrace \pm 1\rbrace ^S\) , while two configurations are adjacent iff they differ at the assignment
of a single vertex, i.e., their Hamming distance is one.
Similarly, any subset \(\Omega _0\subseteq \lbrace \pm 1\rbrace ^S\)  is considered to be connected if the subgraph
induced by \(\Omega _0\)  is connected.
